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← IIT-JEE 2011 · Paper 2
MathematicsInteger+4Question 56 of 60

IIT-JEE 2011 · Paper 2, Mathematics question 56

Let MM be a 3×33\times3 matrix satisfying
M[010]=[−123],M[1−10]=[11−1],andM[111]=[0012].M\begin{bmatrix}0\\1\\0\end{bmatrix}=\begin{bmatrix}-1\\2\\3\end{bmatrix},\quad M\begin{bmatrix}1\\-1\\0\end{bmatrix}=\begin{bmatrix}1\\1\\-1\end{bmatrix},\quad\text{and}\quad M\begin{bmatrix}1\\1\\1\end{bmatrix}=\begin{bmatrix}0\\0\\12\end{bmatrix}.
Then the sum of the diagonal entries of MM is

Answer and worked solution

Official answer: 9

Columns of MM (MeiMe_i is column ii):

  • Me2=(−1,2,3)Me_2=(-1,2,3).
  • Me1=M(e1−e2)+Me2=(1,1,−1)+(−1,2,3)=(0,3,2)Me_1=M(e_1-e_2)+Me_2=(1,1,-1)+(-1,2,3)=(0,3,2).
  • Me3=M(1,1,1)−Me1−Me2=(0,0,12)−(0,3,2)−(−1,2,3)=(1,−5,7)Me_3=M(1,1,1)-Me_1-Me_2=(0,0,12)-(0,3,2)-(-1,2,3)=(1,-5,7).

Trace. M11+M22+M33=0+2+7=9M_{11}+M_{22}+M_{33}=0+2+7=9.

Answer: 9

Topics:Matrix algebra
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